Inequality for square summable complex series ★★

Author(s): Retkes

\begin{conjecture} For all $\alpha=(\alpha_1,\alpha_2,\ldots)\in l_2(\cal{C})$ the following inequality holds $$\sum_{n\geq 1}|\alpha_n|^2\geq \frac{6}{\pi^2}\sum_{k\geq0}\bigg| \sum_{l\geq0}\frac{1}{l+1}\alpha_{2^k(2l+1)}\bigg|^2 $$


Keywords: Inequality

Inequality of the means ★★★


\begin{question} Is is possible to pack $n^n$ rectangular $n$-dimensional boxes each of which has side lengths $a_1,a_2,\ldots,a_n$ inside an $n$-dimensional cube with side length $a_1 + a_2 + \ldots a_n$? \end{question}

Keywords: arithmetic mean; geometric mean; Inequality; packing

Sums of independent random variables with unbounded variance ★★

Author(s): Feige

\begin{conjecture} If $X_1, \dotsc, X_n \geq 0$ are independent random variables with $\mathbb{E}[X_i] \leq \mu$, then $$\mathrm{Pr} \left( \sum X_i - \mathbb{E} \left[ \sum X_i \right ] < \delta \mu \right) \geq \min \left ( (1 + \delta)^{-1} \delta, e^{-1} \right).$$ \end{conjecture}

Keywords: Inequality; Probability Theory; randomness in TCS

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